FIG. 1a illustrates a model of a single photovoltaic (PV) cell. The single PV cell can be modeled as a current source Isc 11 in parallel with a forward biased diode Dcell 14 along with a series and a shunt resistance Rs 18 and Rsh 16. A current/sc 11 is proportional to the solar irradiance received by the PV cell and it is also approximately equal to the short circuit current of the panel. A current Id 12 flows through Del 14 while a current I 22 flows through Rs 18.
Rsh 16 represents the slope of the I-V characteristics near short circuit and Rs 18 represents the slope of its I-V characteristics near an open circuit, as is illustrated by curves 24 and 26 of FIG. 1b. Curve 26 illustrates the I-V characteristics of a real PV cell while curve 24 illustrates the I-V characteristics of an ideal PV cell.
The diode determines the non-linear (exponential) course of the current curve. The PV cell output current can be represented as:I=Isc−Io*{exp(vd/(n−vt))
Where Isc and Io relate to their respective current densities Jsc and Is as follows:Isc=A*Jsc*% irradIo=A*Is. 
Where:
Jsc is the PV cell short circuit density (ma/cm2) at standard (AM 1.5 G, 1000 W/m2, T=25° C.) conditions.
A is the cell area (cm2).
Is is the cell diode reverse bias saturation current.
Vd is the voltage across the cell diode which is in good approximation to the cell voltage V.
n is the emission coefficient for the cell diode. It depends on the manufacturing process (e.g. between 1 and 2).
Vt is the cell diode thermal voltage, in some cells about 26 mv at 300K.
% irrad is the percentage of the sun irradiance in comparison to 1000 W/m2 (100%).
Partial Shading Problems in PV Systems
PV systems can include many PV cells, some of which can be shaded or partially shaded during certain points in time. The shading reduces the overall power generated by a PV system due to the loss of available irradiation caused by shading and due to the so-called mismatch loss caused by partial shading.
Usually a PV system includes many PV cells. An output voltage of a PV cell is determined by its forward biased diode voltage (which can be about 0.55 v but varies depending on the PV cell technology). Because PV systems can be required to output an output voltage of few hundred volts a very large number of PV cells are sometimes connected in a serial manner to provide such an output voltage. For example, if the PV system generates energy that should be provided to a 220 v alternating current (AC) grid then the PV system should generate a direct current (DC) voltage of about 300V or more. This requires serially connecting a few hundred (e.g., about 600) PV cells to each other.
It is noted that PV cells can also be connected in parallel—in order to increase the current/power of the PV system.
Typically long PV cells are “split” to separate PV panels, each including at least one string of PV cells.
When serially connected, a PV cell that is partially or fully shaded reduces the current that flows through the entire string of PV cells—thus reducing the power that can be generated by other PV cells in the string. PV cells that do not generate any current (for example, if they are fully shaded) nullify the power output of the entire serially connected PV cell system.
One known solution to the problem of fully shaded PV cells includes bypass diodes—if a PV cell does not generate adequate power it is bypassed by a bypass diode that is connected in parallel to the PC cell.
FIG. 2 illustrates an equivalent circuit that represents an array of sixteen sequentially connected PV cells. Each PV cells is represented by a PV cell model of a diode (such as D1 22(1), D2 22(2)) that is connected in parallel to a current source (such as 21(1), 21(2)). The current generated by each PV cell is indicated in the figure. The first fourteen PV cells generate 1 A each, while two (fully shaded PV cells) generate 0 A each. These two fully shaded PV cells are represented by PV cell models 20(1) and 20(16) that are bypassed by bypass diode 24.
The model of FIG. 2 also illustrates a load 26 that is connected to the sequence of 16 PV cells and a current lload 28 that passes through the load.
Curve 32 of FIG. 3 illustrates current lload 28 and curve 40 of FIG. 3 illustrates the power (Pload) supplied to the load as a function of the voltage drop (Vload) on the load.
The maximal power is 7 w, assuming that lload 28 equals 1 A, the voltage the falls on each diode is 0.5 volt and each of the diodes are ideal. Fourteen non-shaded serially connected PV cells generate 14*0.4 volts*1 A=7 w.
FIG. 4 illustrates an equivalent circuit that represents an array of sixteen sequentially connected PV cells. Each PV cells is represented by a PV cell model of a diode (such as D1 22(1), D2 22(2)) that is connected in parallel to a current source (such as 21(1), 21(2)). The current generated by each PV cell is indicated in the figure.
The first four PV cells (represented by models 20(1)-20(4)) are partially shaded (30%) and generate 0.7 A each, the last four PV cells (represented by models 20(13)—20(16)) are partially shaded (70%) and 0.3 A each, while other PV cells generate 1 A each.
The first four partially shaded PV cells are bypassed by bypass diode 24(1). The last four partially shaded PV cells are bypassed by bypass diode 24(2).
The model of FIG. 4 also illustrates load 26 that is connected to the sequence of 16 PV cells, a current lload 28 that passes through the load and a voltage Vload 27 developed on load 26.
Curve 54 of FIG. 5 illustrates the without any bypass the 70% shaded PV so cells force a current of 0.3 A through each PV cell. This low current allows a maximal power production of 2.4 W—as illustrated by peak 59 of curve 52.
Curve 53 illustrates lload 28 when neither one of the bypass diodes is connected (and lload=0.3 A), when only bypass diode 24(1) is connected (and lload=0.7 A) and when both bypass diodes are activated to bypass all diodes is (and lload=1 A).
Curve 51 illustrates the power that can be generated by the sequence of PV cells (a) when neither one of the bypass diodes is connected (and lload=0.3 A), (b) when only bypass diode 24(1) is connected (and lload=0.7 A) and (c) when both bypass diodes are activated to bypass all diodes (and lload=1 A).
Curve 51 of FIG. 5 illustrates the power (Pload) supplied to the load as a function of the voltage drop (Vload) 27 on the load. The maximal power production (a) when neither one of the bypass diodes is connected is 2.4 W (peak 59), (b) when only bypass diode 24(1) is connected is 2.4 W (peak 58), (c) when both bypass diodes are activated is 4 W (peak 57.
Curve 51 illustrates that there is a tradeoff between the number of bypassed PV cells and the current generated by each PV cell.
As illustrated above, the bypass prevents that PV cell from reducing the power provided by other serially connected PV cells. On the other hand, the power generated by the bypassed cell does not contribute to the total power generated by the serially connected PV cells.
Another disadvantage of bypassing PV cells is that the output power curve of the group of PV cells has several maximal power peaks (MPP). These peaks can cause mismatches between different PV panels that are connected to each other.